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We consider the placement, delivery promise, and fulfillment decisions of an online retailer. We have a set of products with given numbers of units to be placed at capacitated fulfillment centers. Once we make the placement decisions, we face demands for the products arriving from different demand regions randomly over time. In response to each demand, we pick a delivery promise to offer, which determines the probability that the demand converts into sales as well as choose a fulfillment center to use to serve the demand. Our goal is to decide where to place the units at the beginning of the selling horizon and to find a policy to make delivery promise and fulfillment decisions over the selling horizon so that we maximize the total expected profit. We give an approximation strategy to obtain solutions with performance guarantees for this joint placement, delivery promise, and fulfillment problem. In our approximation strategy, we construct a bounding function that upper bounds the total expected profit from the delivery promise and fulfillment policy when viewed as a function of the placement decisions. To make the placement decisions, we maximize the bounding function subject to the capacity constraints at the fulfillment centers. To make the delivery promise and fulfillment decisions, we construct a policy that obtains a constant fraction of the bounding function. Using our approximation strategy with appropriate bounding functions, we obtain a solution with a constant factor performance guarantee, but if the size of the system, measured by the numbers of units that we need to place and capacities of the fulfillment centers, is large, then we get an asymptotically optimal solution. We compare our approximation strategy with approaches that ignore the interactions between the placement, delivery promise, and fulfillment decisions as well as heuristics that are based on Lagrangian relaxation, demonstrating that our approximation strategy compares quite favorably. 

Jointly Making Inventory Stocking and Assortment Offering Decisions Making the product assortment offer decisions for a customer requires keeping a balance between offering an assortment that will satisfy the current customer and reserving the products with scarce inventories for the customers who will arrive in the future. Therefore, whereas the assortment offer decisions should depend on the current inventories of the products, the stocking decisions should anticipate how the offered assortments will deplete the inventories, thereby creating a natural interaction between inventory stocking and assortment offer decisions. In “Coordinated Inventory Stocking and Assortment Customization,” Bai, El Housni, Rusmevichientong, and Topaloglu develop models that jointly make the assortment offer and inventory stocking decisions. Their models allocate the limited stocking capacity to the inventories for different products, while taking into consideration the assortment offer decisions that will be made for the customers arriving over a selling horizon.